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locally_finite_topology_subset [2016/09/14 15:31] nikolaj |
locally_finite_topology_subset [2016/09/15 01:42] (current) nikolaj |
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| ==== Set ==== | ==== Set ==== | ||
| | @#55CCEE: context | @#55CCEE: $\langle X,\mathcal T\rangle$ ... topological space | | | @#55CCEE: context | @#55CCEE: $\langle X,\mathcal T\rangle$ ... topological space | | ||
| - | | @#FFBB00: definiendum | @#FFBB00: $C$ in it | | + | | @#FFBB00: definiendum | @#FFBB00: ${\mathcal C} in it | |
| - | | @#AAFFAA: inclusion | @#AAFFAA: $C\subset \mathcal T$ | | + | | @#AAFFAA: inclusion | @#AAFFAA: ${\mathcal C}\subset \mathcal T$ | |
| | @#FFFDDD: for all | @#FFFDDD: $x\in X$ | | | @#FFFDDD: for all | @#FFFDDD: $x\in X$ | | ||
| | @#FFFDDD: exists | @#FFFDDD: $V\in \mathcal T$ | | | @#FFFDDD: exists | @#FFFDDD: $V\in \mathcal T$ | | ||
| | @#55EE55: postulate | @#55EE55: $x\in V$ | | | @#55EE55: postulate | @#55EE55: $x\in V$ | | ||
| - | | @#55EE55: postulate | @#55EE55: $\{U\in C\,|\,U\cap V\neq\emptyset\}$ ... finite | | + | | @#55EE55: postulate | @#55EE55: $\{U\in {\mathcal C}\,|\,U\cap V\neq\emptyset\}$ ... finite | |
| ----- | ----- | ||
| === Idea === | === Idea === | ||
| - | Like many properties, this is a notion of smallness, and more particularly, not smallness of a subset of $X$ but smallness of a collection $C$ of subsets of $X$. | + | Like many properties, this is a notion of smallness. It's not about the smallness of a subset $U$ of $X$, but smallness of a collection ${\mathcal C}$ of subsets $U$ of $X$. |
| - | You choose a small sample of neighborhoods (the sets $V\in\mathcal T$) and $C$ ought to be countable w.r.t. to that sample, i.e. pro $V$. | + | You may consider a well choosen sample of neighborhoods (the sets $V\in{\mathcal T}$) and ${\mathcal C}$ ought to be finite with respect to that sample (finite //pro// $V$). |
| === Dicussion === | === Dicussion === | ||
| - | A topologal space is paracompact if it has a cover with that property. | + | * A topologal space is //paracompact// if it any cover has a refinement with that property. |
| + | * The sample of $V$'s above may be very big, so ${\mathcal C}$ is really only small w.r.t. the sample. In a //compact// space, on the other hand, the cover itself is finite (and you don't need to consider that sample). | ||
| + | * Note that the name //locally compact// is already used for the situation where every point $x\in X$ has a compact neighborhood $V$. | ||
| === Reference === | === Reference === | ||
